New spectral criteria for almost periodic solutions of evolution equations (Q2717567)

The authors consider the linear, inhomogeneous integral equation NEWLINE\[NEWLINEx(t)= U(t,s) x(s)+ \int^t_s U(t,\xi) f(\xi) d\xi,\tag{1}NEWLINE\]NEWLINE where \(f\) is continuous and \((U(t, s))_{t\geq s}\) is a 1-periodic evolutionary process in a complex Banach space. This problem yields results for the problem NEWLINE\[NEWLINE{dx\over dt}= A(t) x+ f(t),\tag{2}NEWLINE\]NEWLINE where \(A\) is the generator of a \(C_0\)-semigroup and \(f\) is bounded and uniformly continuous with precompact range. The authors prove a spectral decomposition result for bounded uniformly continuous solutions to (1). This theorem is then used to prove the existence of almost-periodic (in the sense of Bohr) solutions with specific spectral properties. A corollary specifies the assumptions, including that \(f\) be almost-periodic, so that if there exists a bounded uniformly continuous solution \(u\) to (1), then there exists an almost-periodic solution \(w\) to (1) such that \(\sigma(w)= \sigma(f)\), where \(\sigma(w)\) is defined to be the closure of \(\{e^{i\xi}: \xi\in \text{sp}(w)\}\) and sp represents the Carleman spectrum. Another corollary specifies the assumptions so that if there exists a bounded uniformly continuous mild solution to (2), then there exists a quasi-periodic mild solution \(w\) with \(\text{sp}(w)= \text{sp}(f)\). The authors close the paper with a number of examples.